unit – The ball must always move towards the poles of the planet

I have a ball (player) that moves around the planet (also a ball, but larger). The player must travel the planet passing circles from pole to pole. It has the ability to move left or right in order to avoid obstacles. How to focus the ball on the posts? I don't know how to make the player move but not bow out of the desired course.

I have a ball to which I apply force

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And its child object to which is attached the script where the direction of the force applied to the ball is taken

public class PlanetConstraint : MonoBehaviour
{
    (SerializeField) private Transform _targetPlanet;

    private void FixedUpdate()
    {
        Quaternion rotation = Quaternion.FromToRotation(-transform.up, _targetPlanet.position - transform.position);
        transform.rotation = rotation * transform.rotation;
    }
}

I think it is worth turning the child around the transform.up axis so that transform.forward is facing the pole.

More information:

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functions – Catenary: Distance between poles without pole height

I looked at some catenary responses and they are great!
i have a little problem with mine however ..
i need to find the distance between the poles AND the minimum height of said poles.
now the information I have is:
Cable length = 15 m,
middle and lowest cable height: 5.6 m,
the posts must have the same height …

if anyone could point me in the right direction, i would be super grateful!

Thank you.

complex analysis – Concerning a doubt in the calculation of the poles of the Laurent series linked to the cot (πz)

When studying the analytic number theory in Tom M Apostol's book, I couldn't think of how to calculate the poles of this function.

The function is – $ F_n (z) $ =
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The main problem I face is that I don't know how to find the expansion of the Laurent series from cot (πz).

Can anyone tell how to find their poles?

Aggressive geometry – When considering poles for rational functions in a Riemann compact surface, how do you calculate the number of poles allowed?

I work in an algebra of rational functions in compact Riemann surfaces with an arbitrary kind. The idea that I fight is how to count the number $ n $ of poles allowed in a Riemann surface like that $ mathbb {C}[t,t^{-1},u]/ langle u ^ m-p (t) rangle $.

I do a thorough research in previous references looking for a formula that could give me that number and I have the impression that it's just a language problem between algebraic-geometry and the papers I'm working with. Am I right?

Sometimes, in my research, the calculation of the authorized poles turned out to be a simple statement that gives me the impression that this could be a fundamental fact in algebraic geometry. I'll give you an example, just a copy of one of the papers I'm dealing with:

Proposal 1: Let $ p (t) = sum_ {i in mathbb {Z}} a_it ^ i in mathbb {C}
$ R = mathbb {C}[t,t^{-1},u]/ langle u ^ 2-p (t) rangle $.

The number $ n $ where the poles are allowed $ R $ depend on $ p (t) $
according to the formula $ n = 4-r $ or $ r $ is the number of ramified
points in $ {0, infty } $: $ 0 is branched exactly when the constant
term $ a_0 = $ 0, and $ infty $ is branched exactly when the degree $ d $ is
odd.

Even opening all the references, I do not find some clearly explaining how to achieve Proposition 1 (which I call the hyperelliptic case). My research institutes working in algebra do not know how to find it either.

I would like to know how to count the poles allowed in $ mathbb {C}[t,t^{-1},u]/ langle u ^ m-p (t) rangle $ (which I call the superelliptic case). Is there a formula?

Ag.algebraic geometry – How to determine the kind and number of poles in a ring $ mathbb {C}[t,t^{-1},u]/ langle u ^ m-p (t) rangle $?

A pole of the finite polynomial $ p (t) = sum_ {i in mathbb {Z}} a_it ^ i $ is a point $ p_0 $ such as $ p (p_0) = infty $.

I already know that when $ m = $ 2, genre $ g $ depends on the definition polynomial $ p (t) $ with degree $ d $ according to the formula
$$
g = frac {d-1} {2},
$$

in an equivalent way
$$
2g = begin {case} d-2 textrm {if $ d $ is even} \ d-1 textrm {if $ d $ is odd.} End {cases}
$$

After that, I have that number $ n $ where the poles are allowed depends on $ p (t) $ according to the formula $ n = 4-r $ or $ r $ is the number of branched points in $ {0, infty } $: $ 0 is branched exactly when the term constant $ a_0 = $ 0, and $ infty $ is branched exactly when the degree $ d $ is odd. The combination of this information gives
$$
2g + n-1 = start {observations} d + 1 textrm {if} a_0 neq0, \ d-1 textrm {if} a_0 = 0. end {cases}
$$

As I said, that's the case when $ m = $ 2. When $ m $ is arbitrary, what can I say?

How to determine the kind and number of poles in a ring
$ mathbb {C}[t,t^{-1},u]/ langle u ^ m-p (t) rangle $?

Function greens – How to solve this integral complex using poles?

I want to find the function of green of a free particle, which depends on the integral:
$$
I = frac {1} {4 piąir} int ^ {+ infty} _ {- infty} frac {ke ^ {ikr}} {E- frac { hbar²k²} {2m} + i eta} dk ,.
$$

Then, using the main value of Cauchy, we remove the $ eta $… The result is as follows:

$$
###
$$